Statistical Features of Gaussian/Normal Distribution CurvePresented To: Dr. Islam Ullah Khan

Presented By: Khadeeja IkramRoll No: 0164-BH-CHEM-11

Gaussian Distribution The normal (or Gaussian) distribution is

a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.

The normal distribution is sometimes informally called the bell curve. However, many other distributions are bell-shaped.

The terms Gaussian function and Gaussian bell curve are also ambiguous because they sometimes refer to multiples of the normal distribution that cannot be directly interpreted in terms of probabilities.

IntroductionThe probability density of the normal distribution

is:

Here, µ is the mean or expectation of the distribution (and also its median and mode). The parameter σ

 is its standard deviation; its variance is then  σ 2. A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.

If µ= 0 and σ= 1 , the distribution is called the standard normal distribution or the unit normal distribution denoted by N(0,1) and a random variable with that distribution is a standard normal deviate.

History of Gaussian Curve

The Normal Probability distribution, which is considered the cornerstone of the modern statistical theory, was discovered by Abraham de Moivre as the limiting form of the binomial distribution by increasing n, the number of trials, to a very large number for a fixed value of P.

The name of Pierre S. Laplace is also associated with the derivation of the normal distribution.

The normal distribution is also called the Gaussian distribution in the honour of the great German mathematician Carl F. Gauss , who also derived its equation mathematically as the probability distribution of the errors measurements.

He was karl Pearson who in 1893 called it the normal distribution and is the best known by this name today.

Standardized Normal DistributionA normal probability distribution depends on

the values of the parameters µ, and σ 2 the various possible values for these two parameters will result in an unlimited number of different normal distribution.

Mathematical Form

The Z= (X - μ) / σ has zero mean and unit variance. Every normally distributed X with mean= µ and variance= σ 2 is therefore conventionally transformed into a normal Z with zero mean and unit variance by using the following expression

Z= (X - μ) / σ

Features of Gaussian Distribution CurveThe main features of the normal distribution

are given below: A normal distribution is bell-shaped

(symmetric).

The mean, median, and mode are equal and are located at the center of the distribution.

The function f(x) defining the normal distribution is a proper p.d.f. (probability density function), i.e. f(x)≥0 and the total area under the normal curve is unity.

The mean and variance of the normal distribution are µ and σ 2 respectively.

The mean deviation of the normal distribution is approximately 4/5 of its standard deviation.

The normal curve has points of inflection which are equidistant from the mean.

For the normal distribution, the odd order moments about the mean are all zero and the even order moments about the mean are given by µ2n = (2n-1) (2n-3)… 5.3.1 σ2n

The sum of the independent normal variables is a normal variable.

If X is N (µ, σ 2) and if Y=a+bX, then Y is N (a+bµ, b2 σ2).

No matter what the values of µ and σ are, areas under normal curve remain in certain fixed proportions with the specified number of standard deviations on either side of µ. For example, the interval

I. Approximately 68% of the area under the curve is between µ-σ and µ+σ.

II. Approximately 95% of the area under the curve is between µ-2σ and µ+2σ.

III. Approximately 99.7% of the area under the curve is between µ-3σ and µ+3σ.

The normal curve approaches, but never really touches , the horizontal axis on either side of the mean towards plus and minus infinity, that is the curve is asymptotic to the horizontal axis as x ±∞.

References Chaudhary, M. S. and Kamal, S., Introduction Statistical Theory, part 1,

Ilmi Kitab Khana Urdu Bazar Lahore. (2002) http://highered.mheducation.com/sites/dl/free/0073521485/940175/doane4

e_sample_ch07.pdf http://www.amsi.org.au/ESA_Senior_Years/PDF/ContProbDist4e.pdf http://trojan.troy.edu/studentsupportservices/assets/documents/presentatio

ns/math_science/Math_2200_Sections_6_2-6_3.pptx http://www.cliffsnotes.com/math/statistics/sampling/properties-of-the-norm

al-curve http://www.mathnstuff.com/math/spoken/here/2class/90/normal.htm http://s3.amazonaws.com/ppt-download/normaldistribution-110307043247-

phpapp01.pptx?response-content-disposition=attachment&Signature=PZziVFZj82%2FJI1iyMxl%2BREtYKLQ%3D&Expires=1431022477&AWSAccessKeyId=AKIAIA7QTBOH2LDUZRTQ